Given a parametric polynomial curve $\gamma:[a,b]\rightarrow \mathbb{R}^n$, how can we sample a random point $\mathfrak{x}\in \mathrm{im}(\gamma)$ in such a way that it is distributed uniformly with respect to the arc-length? Unfortunately, we cannot sample exactly such a point-even assuming we can perform exact arithmetic operations. So we end up with the following question: how does the method we choose affect the quality of the approximate sample we obtain? In practice, there are many answers. However, in theory, there are still gaps in our understanding. In this paper, we address this question from the point of view of complexity theory, providing bounds in terms of the size of the desired error.

翻译：考虑到一个参数多角度曲线$\gamma:[a,b]\rightrow \mathbb{R ⁇ n$,我们如何对随机点 $\mathfrak{x ⁇ in\mathrm{im}(\gamma}) 进行抽样?不幸的是,即使假设我们能够进行精确的算术操作,我们也不能对这个点进行精确的抽样抽样。所以我们最后要回答以下问题:我们选择的方法如何影响我们获得的近似样本的质量?实际上,有很多答案。然而,理论上,我们的理解仍然存在差距。在本文中,我们从复杂理论的角度来处理这个问题,提供了理想错误大小的界限。